Solve the equation. $\dfrac{dy}{dx}=\dfrac{4x^3}{\cos(y)}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\arccos(4x^3)+C$ (Choice B) B $y=\arccos(4x^3+C)$ (Choice C) C $y=\arcsin(x^4+C)$ (Choice D) D $y=\arcsin(x^4)+C$
We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{4x^3}{\cos(y)} \\\\ \cos(y)\,dy&=4x^3\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \cos(y)\,dy&=4x^3\,dx \\\\ \int \cos(y)\,dy&=\int 4x^3\,dx \\\\ \sin(y)&=x^4+C \\\\ \arcsin(\sin(y))&=\arcsin(x^4+C) \\\\ y&=\arcsin(x^4+C) \end{aligned}$ Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\arcsin(x^4+C)$